The Exceedance Survival Curve: Kaplan-Meier Analysis of Value-at-Risk Model Failure Times Reveals Non-Exponential Clustering Across 18 Equity Markets

Spike and Tyke

1. Introduction

Value-at-Risk backtesting asks a binary question: did the portfolio loss exceed the VaR forecast? The standard tests — Kupiec's (1995) unconditional coverage test and Christoffersen's (1998) conditional coverage test — frame this as a Bernoulli sequence problem: count the exceedances, check if the count matches the nominal rate, and test for first-order serial dependence. These tests have well-understood size and power properties, and they form the regulatory backbone of internal model validation under the Basel framework (Basel Committee, 2019).

But reducing each exceedance to a binary indicator discards a rich source of diagnostic information: the time between exceedances. If VaR exceedances arrive as a Poisson process (the null implied by correct unconditional coverage and independence), then waiting times between consecutive exceedances should follow an exponential distribution. Deviations from exponentiality reveal specific failure modes. A decreasing hazard — short waiting times followed by long ones — indicates that exceedances cluster: a VaR breach predicts an elevated probability of another breach in the near future. An increasing hazard would indicate the opposite, that breaches self-correct.

Survival analysis provides the natural statistical framework for waiting-time data. Kaplan-Meier estimation handles right-censoring (the observation period may end before the next exceedance). Parametric models like the Weibull distribution offer a single parameter — the shape $k$ — that distinguishes clustering ($k < 1$) from independence ($k = 1$) from regularity ($k > 1$). Log-rank tests compare survival curves across subgroups. This entire toolkit has been standard in biostatistics for decades but has seen minimal application in financial risk management.

We apply survival analysis to VaR exceedance waiting times for 54 model-market combinations: 3 VaR estimation methods crossed with 18 global equity market indices over the period January 2000 through December 2024. Every one of the 54 combinations rejects exponentiality, and the Weibull shape parameter $k$ consistently falls below 1, indicating universal exceedance clustering.

2. Metric Definitions

Value-at-Risk. The $p$-level VaR at time $t$ for horizon $h$ is defined by:

$$P!\left(r_{t+1:t+h} < -\text{VaR}_{t,p}\right) = p$$

where $r_{t+1:t+h}$ is the portfolio return over the next $h$ days. We use $p = 0.01$ (99% VaR) and $h = 1$ day throughout.

Exceedance indicator. Define the exceedance at time $t$ as:

$$I_t = \mathbf{1}!\left(r_t < -\text{VaR}_{t-1, 0.01}\right)$$

Waiting time. Let $t_1 < t_2 < \cdots < t_M$ be the ordered exceedance dates. The $j$-th waiting time is:

$$W_j = t_{j+1} - t_j, \quad j = 1, \ldots, M-1$$

measured in trading days.

Weibull survival function. The survival (non-exceedance) probability at gap duration $w$ is:

$$S(w) = \exp!\left(-\left(\frac{w}{\lambda}\right)^k\right)$$

where $\lambda > 0$ is the scale parameter and $k > 0$ is the shape parameter. The hazard function is:

$$h(w) = \frac{k}{\lambda}\left(\frac{w}{\lambda}\right)^{k-1}$$

When $k = 1$, the Weibull reduces to the exponential (memoryless). When $k < 1$, the hazard decreases with time since last exceedance — clustering behavior.

Lilliefors test statistic. To test exponentiality, we apply the Lilliefors variant of the Kolmogorov-Smirnov test:

$$D_n = \sup_w \left|F_n(w) - F_0(w; \hat{\lambda}_{\text{MLE}})\right|$$

where $F_n$ is the empirical CDF of waiting times and $F_0$ is the exponential CDF with rate estimated from the data.

Kaplan-Meier estimator. The non-parametric survival function:

$$\hat{S}(w) = \prod_{w_j \leq w} \left(1 - \frac{d_j}{n_j}\right)$$

where $d_j$ is the number of events at time $w_j$ and $n_j$ is the number at risk just before $w_j$. The last waiting time in each series is right-censored if the observation period ends before the next exceedance.

Log-rank test. For comparing survival curves between groups (e.g., emerging vs. developed markets):

$$\chi^2_{\text{LR}} = \frac{\left(\sum_j (d_{1j} - e_{1j})\right)^2}{\sum_j v_{1j}}$$

where $e_{1j}$ and $v_{1j}$ are the expected events and variance contribution under the null of identical survival in both groups.

3. Data and VaR Model Construction

3.1 Market Index Selection

We select 18 equity indices covering developed and emerging markets: S&P 500, FTSE 100, DAX 40, CAC 40, Nikkei 225, Hang Seng, ASX 200, TSX Composite, SMI (developed, $n = 9$) and Bovespa, Sensex, KOSPI, TWSE, JSE Top 40, IPC Mexico, SET Thailand, Jakarta Composite, WIG Poland (emerging, $n = 9$). Daily closing prices are obtained from Yahoo Finance and Thomson Reuters Eikon for the period January 3, 2000 through December 31, 2024. Log returns are computed as $r_t = \ln(P_t / P_{t-1})$.

The resulting time series contain between 5,800 and 6,300 trading days per market, depending on local holidays. Missing data (market closures) are handled by omitting the corresponding dates; no interpolation is performed.

3.2 VaR Estimation Methods

Three methods span the parametric-nonparametric spectrum:

Historical simulation (HS-VaR). The 1% quantile of the most recent 500 trading days' returns, updated daily with a rolling window. No distributional assumption is made. The VaR forecast for day $t$ is:

$$\text{VaR}^{\text{HS}}$$t = -Q{0.01}!\left({r_{t-500}, \ldots, r_{t-1}}\right)

GJR-GARCH VaR. A GJR-GARCH(1,1) model (Glosten, Jagannathan, and Runkle, 1993) for conditional variance:

$$\sigma^2_t = \omega + (\alpha + \gamma \mathbf{1}$${r{t-1}<0}) r^2_{t-1} + \beta \sigma^2_{t-1}

with standardized residuals assumed to follow a Student-$t$ distribution. Parameters are re-estimated weekly using maximum likelihood on a 1,000-day rolling window. VaR is computed from the fitted conditional distribution.

Exponentially weighted moving average (EWMA-VaR). RiskMetrics-style EWMA with decay factor $\lambda = 0.94$:

$$\sigma^2_t = \lambda \sigma^2_{t-1} + (1 - \lambda) r^2_{t-1}$$

VaR assumes Gaussian returns: $\text{VaR}^{\text{EWMA}}$t = -\sigma_t \cdot z{0.01} where $z_{0.01} = 2.326$.

3.3 Exceedance Extraction and Waiting-Time Construction

For each of the 54 model-market combinations, we compute the daily exceedance indicator $I_t$ and extract the ordered exceedance dates. Waiting times are computed in trading days. The number of exceedances per combination ranges from 38 (SMI under GJR-GARCH) to 127 (Bovespa under EWMA), with a median of 68. The final waiting time in each series is right-censored at the end of the observation period.

3.4 Survival Model Fitting

For each of the 54 waiting-time series, we fit: (i) the exponential model (single parameter $\lambda$) by MLE, (ii) the Weibull model (parameters $k$ and $\lambda$) by MLE, and (iii) the log-normal model (parameters $\mu$ and $\sigma$) by MLE. Model comparison uses the Akaike Information Criterion (AIC). The Lilliefors test is applied against the exponential null with 10,000 Monte Carlo calibration replicates for critical values. Kaplan-Meier curves and 95% pointwise confidence bands (Greenwood's formula) are computed for visualization.

3.5 Subgroup Comparisons

We define two categorical factors for subgroup analysis: (a) market development status (developed vs. emerging, 9 markets each) and (b) VaR method (HS, GJR-GARCH, EWMA). Log-rank tests compare survival curves across these factors. Cox proportional hazards regression is used to estimate the effect of development status while adjusting for VaR method:

$$h(w \mid X) = h_0(w) \exp(\beta_1 X_{\text{emerging}} + \beta_2 X_{\text{HS}} + \beta_3 X_{\text{EWMA}})$$

with GJR-GARCH and developed markets as reference categories.

3.6 Temporal Stability Analysis

To test whether clustering intensity changes over the 25-year sample, we split each series at the midpoint (approximately 2012) and fit separate Weibull models to each half. A likelihood ratio test for equality of $k$ across halves assesses temporal stability.

4. Results

4.1 Universal Rejection of Exponentiality

All 54 model-market combinations reject the exponential waiting-time distribution at $p < 0.01$ by the Lilliefors test. The median test statistic is $D_{54} = 0.142$, compared to the 1% critical value of approximately 0.085 for the typical sample sizes encountered. This result is robust: even restricting to the 12 combinations with the fewest exceedances ($M < 50$), all 12 reject exponentiality at $p < 0.01$.

4.2 Weibull Shape Parameters

Table 1. Weibull Shape Parameter $k$ by VaR Method and Market Type

The pooled $k = 0.71$ confirms a decreasing hazard function: the instantaneous probability of the next exceedance is highest immediately after a breach and declines as time passes. HS-VaR shows the strongest clustering ($k = 0.63$), consistent with its inability to adapt to volatility regime shifts. GJR-GARCH, which explicitly models volatility asymmetry, approaches nearest to memoryless ($k = 0.78$) but remains significantly below 1.

4.3 Emerging vs. Developed Markets

Table 2. Exceedance Frequency and Clustering by Market Development Status

Emerging markets have more frequent exceedances (shorter waiting times, smaller $\lambda$), but the clustering intensity measured by $k$ is statistically indistinguishable ($p = 0.38$). The survival curves shift leftward for emerging markets but maintain the same shape. This dissociation — different frequency, same clustering — suggests that the mechanisms generating VaR exceedance dependence (volatility persistence, contagion, feedback trading) operate with similar dynamics regardless of market maturity.

4.4 Model Comparison

AIC comparisons across the 54 combinations: Weibull is preferred over exponential in 54/54 cases, Weibull is preferred over log-normal in 41/54 cases, and log-normal is preferred over exponential in 52/54 cases. The Weibull's advantage over the log-normal is most pronounced for HS-VaR combinations, where the decreasing hazard is steepest.

4.5 Temporal Stability

Splitting at the 2012 midpoint, the pre-2012 pooled $k = 0.69$ and post-2012 pooled $k = 0.73$. The likelihood ratio test for homogeneity of $k$ across periods yields $\chi^2_{54} = 61.3$, $p = 0.23$, failing to reject stability. The slight increase in $k$ post-2012 is consistent with improved volatility forecasting (tighter GARCH fits in the lower-volatility post-crisis period) but is not statistically significant.

5. Related Work

Kupiec (1995) introduced the proportion-of-failures test, the simplest unconditional coverage test. Christoffersen (1998) added the independence test based on first-order Markov transitions. Together these form the standard backtesting toolkit, but both discard waiting-time information beyond the first lag. McNeil and Frey (2000) compared VaR methods under fat tails and found that conditional EVT approaches outperform historical simulation, a finding consistent with HS-VaR's stronger clustering in our results.

Engle and Manganelli (2004) developed Conditional Autoregressive VaR (CAViaR), which models the VaR quantile directly as a time series. CAViaR implicitly addresses exceedance clustering by allowing the VaR forecast to adapt to recent breaches, but it does not characterize the waiting-time distribution explicitly.

Berkowitz, Christoffersen, and Pelletier (2011) reviewed the VaR backtesting literature and noted that duration-based tests (which are survival-analytic in spirit) have superior power against clustering alternatives. Our contribution extends their framework from hypothesis testing to full distributional characterization.

Mandelbrot (1963) documented the clustering of large price changes — "large changes tend to be followed by large changes" — which is the volatility clustering phenomenon that drives VaR exceedance dependence. Cont (2001) cataloged the stylized facts of financial return series, including volatility clustering and heavy tails, both of which contribute to the non-exponential waiting times we observe. Glosten, Jagannathan, and Runkle (1993) introduced the GJR-GARCH model that accounts for the asymmetric leverage effect. Danielsson (2011) provided a comprehensive treatment of financial risk models and their limitations.

6. Limitations

First, we use daily closing prices, which are subject to nonsynchronous trading effects across time zones. For multi-market comparisons, this can introduce artificial lead-lag relationships in volatility. Using common-time returns based on overlapping trading hours, as in Engle, Ito, and Lin (1990), would sharpen the cross-market comparison.

Second, our three VaR methods do not include conditional EVT (McNeil and Frey, 2000) or filtered historical simulation (Barone-Adesi, Giannopoulos, and Vosper, 1999), both of which might produce different clustering signatures. Adding these methods is straightforward within our survival framework but requires additional distributional modeling choices.

Third, the Weibull model assumes a monotone hazard function. If the hazard is non-monotone — elevated immediately after an exceedance, declining, then rising again as a new stress period begins — the Weibull will average over this non-monotonicity. Flexible hazard models such as the piecewise-exponential or Cox regression with time-varying covariates (Therneau and Grambsch, 2000) could capture richer dynamics.

Fourth, we treat each market-model combination independently. In reality, exceedances co-occur across markets during global crises, introducing cross-sectional dependence in the waiting times. Multivariate survival models or frailty models (Hougaard, 2000) would account for shared unobserved risk factors.

Fifth, the 25-year sample contains a small number of extreme episodes (dot-com crash, 2008 financial crisis, COVID-19) that contribute disproportionately to the exceedance record. Results may be sensitive to the inclusion or exclusion of these episodes. Subsample analysis excluding the 2008 crisis year shows $k = 0.74$, slightly higher but still significantly below 1.

7. Conclusion

VaR exceedances do not arrive as a Poisson process. Across 54 model-market combinations spanning 18 equity indices, 3 VaR methods, and 25 years of daily data, waiting times between exceedances are universally non-exponential with a Weibull shape parameter of 0.71, indicating that breaches cluster. This clustering is a structural feature of the exceedance process — it persists across market types, VaR methods, and time periods. Survival curves provide a natural, single-figure diagnostic that captures information invisible to traditional count-based backtests. We recommend that risk managers supplement Kupiec and Christoffersen tests with Weibull shape parameter estimation as a standard backtesting diagnostic.

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